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e5a97c6 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 | import Mathlib
import OAI.Combinatorics.MatchingCount.Graphs.TreeMatchingCheck
namespace OAI
namespace MatchingFPRAS.TreeTyped.SparseProgram
open TreeTyped
variable {E V C : Type} [Codec E] [Codec V] [Codec C] [LinearOrder E] [LinearOrder V]
attribute [local instance] Transport.Sparse.finsetOrder
def certify (p : C×(E×V)→Bool) (x : (C×Finset V)×Finset (Finset E)) : Finset (Finset E) :=
x.2.filter (fun a=>matchingCheck p (x.1,a)=true)
def certifyRealizer {p : C×(E×V)→Bool} (P : Realizer p) : Realizer (certify p) :=
(finsetFilterWith (matchingCheckRealizer P)).congr (fun _=>rfl)
lemma polynomial_certify {p : C×(E×V)→Bool} {P : Realizer p} (hP : PolynomialTime P) :
PolynomialTime (certifyRealizer P) :=
polynomial_congr _ (polynomial_finsetFilterWith (polynomial_matchingCheck hP))
variable [Fintype E] [Fintype V] (ends : E→Sym2 V)
attribute [local instance] Transport.Sparse.pmOrder
def pmEmbedding : @OrderEmbedding (Transport.PM (fun e v=>v∈ends e)) (Finset E)
(Transport.Sparse.pmOrder ends).toLE (Transport.Sparse.finsetOrder (E:=E)).toLE :=
{ toFun := Subtype.val
inj' := Subtype.val_injective
map_rel_iff' := Iff.rfl }
omit [Codec E] [Codec V] [Codec C] in
lemma certify_eq (p : C×(E×V)→Bool) (c : C)
(hp : ∀ e v,p (c,(e,v))=decide (v∈ends e)) (s : Finset (Finset E)) :
certify p ((c,Finset.univ),s)=(Transport.Sparse.certify ends s).map (pmEmbedding ends).toEmbedding := by
ext a
simp only [certify,Finset.mem_filter,matchingCheck_eq p c _ hp,decide_eq_true_eq,
Finset.mem_map,pmEmbedding]
constructor
· rintro ⟨ha,hm⟩
exact ⟨⟨a,hm⟩,by simpa using ha,rfl⟩
· rintro ⟨b,hb,rfl⟩
exact ⟨by simpa using hb,b.property⟩
end MatchingFPRAS.TreeTyped.SparseProgram
end OAI
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